Place 3/4 cup papaya, lime juice, rum, and vanilla in a food processor, and process until smooth; set aside.
Combine water, piloncillo, cloves, and cinnamon in a medium saucepan; cook over medium heat 13 minutes. Increase heat to medium-high, and cook an additional 10 minutes or until the consistency of thin syrup (do not stir). Remove from heat; let stand 1 minute. Discard cloves and cinnamon stick. Carefully stir in pureed papaya mixture. Pour sauce into a bowl; set aside.
Thread remaining papaya cubes and pineapple cubes alternately onto each of 6 (10-inch) skewers. Place skewers on broiler pan coated with cooking spray; broil 8 minutes or until lightly browned, basting occasionally with caramel sauce. Serve kebabs with remaining sauce.